Bilbao Crystallographic Server arrow Transformation matrix

The transformation matrix

A general change of the coordinate system involves both an origin shift and a change of the basis and is described by the matrix-column pair (P, p).


For example:

The expression (P, p) = (a-b,a+b,2c; 0,0,1/2) stands for

P =
110
-110
002
0
and p=0
1/2


Euler angles

According to Euler's rotation theorem, any rotation can be discribed using three angles (θ, φ, Ψ).



A general rotation can be written as:

R =
Cosθ-Sinθ0
SinθCosθ0
001
100
0Cosφ-Sinφ
0SinφCosφ
CosΨ0SinΨ
010
-SinΨ0CosΨ




[*] For more information: International Tables for Crystallography. Vol. A, Space Group Symmetry. Ed. Theo Hahn (3rd ed.), Dordrecht, Kluwer Academic Publishers, Section "Transformations in crystallography", 1995.

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